MATH 202 Exam 1, Spring 2005

MATHEMATICS 202 SECOND SEMESTER, 2004-05 QUIZ 1 Time: 55 MINUTES. Date: March 23, 2005. Name: ID Number: Circle Problem Session Instructor: Dr. A.Lyzzaik, Dr. H. Yamani Circle Section Number: 15, 16, 17, 18, 19 1, Solve the initial-value problem (8 points) 2. Solve the differential equation (8 points) 3. Solve the initial value problem (a? + y? + zy) dz — xy dy = 0 y(1) =e-1. (8 points) 4. Find the general solution of the differential equation 4a*y" +y=0 for x > 0 if one of the solutions is y, = z!/? Inz. 5. Find the genera! solution of the differential equation y” + 6y” + y! — 34y = 0. (8 points) 6. Answer TRUE (T) or FALSE (F) only: (a) The set of functions {z?, 2|z}} on ] — 00, 00 is linearly indepen- dent. (2 points) (b) An initial-value problem di - = f(x,y), y(20) = yo has a unique solution if f(x,y) and f, (2 points) (c) (z,y) are continuous in the zy~plane. A set {y1, ya} of solutions of y” +y'+y=00n ] — 00, oof is linearly independent if and only if the Wronskian W(y1,y2) is nonzero at some x €] — 00, oof. (2 points) (d) ——- The initial-value problem (2? + 1)y" + zy’ +y =0y(0) = y'(0) =0 has the unique solution y = 0 on ] — co, oof. (2 points) (e) The differential equation (cosz sinz — xy) dr + y(1— 2?).= 0 is exact.